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Post Closed as "Not suitable for this site" by abx, Mark Grant, Will Sawin, Andreas Blass, user44191
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MacFly
MacFly
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Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that :

  • $X=\sqcup_{i=1}^{n}U_i$
  • the map $x\in X \mapsto \text{Card}(\rho^{-1}(\{x\})$$x\in X \mapsto \text{Card}(\rho^{-1}(\{x\}))$ is constant on $U_i$ for all $1≤i≤n$

I have already proved that the map $x\mapsto \text{Card}(\rho^{-1}(\{x\})$$x\mapsto \text{Card}(\rho^{-1}(\{x\}))$ is continuous on $X$ but i don't know how to continue ... I also thought of a proof by induction on $n=\text{sup}\{\text{Card}(\rho^{-1}(\{x\})),\: x\in X\}$ which is clear for $n=1$.

Thank you for you help

Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that :

  • $X=\sqcup_{i=1}^{n}U_i$
  • the map $x\in X \mapsto \text{Card}(\rho^{-1}(\{x\})$ is constant on $U_i$ for all $1≤i≤n$

I have already proved that the map $x\mapsto \text{Card}(\rho^{-1}(\{x\})$ is continuous on $X$ but i don't know how to continue ... I also thought of a proof by induction on $n=\text{sup}\{\text{Card}(\rho^{-1}(\{x\})),\: x\in X\}$ which is clear for $n=1$.

Thank you for you help

Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that :

  • $X=\sqcup_{i=1}^{n}U_i$
  • the map $x\in X \mapsto \text{Card}(\rho^{-1}(\{x\}))$ is constant on $U_i$ for all $1≤i≤n$

I have already proved that the map $x\mapsto \text{Card}(\rho^{-1}(\{x\}))$ is continuous on $X$ but i don't know how to continue ... I also thought of a proof by induction on $n=\text{sup}\{\text{Card}(\rho^{-1}(\{x\})),\: x\in X\}$ which is clear for $n=1$.

Thank you for you help

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MacFly
MacFly
  • 53
  • 4

Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that :

  • $X=\sqcup_{i=1}^{n}U_i$
  • the map $x\in X \mapsto \text{Card}(\rho^{-1}(\{x\})$ is constant on $U_i$ for all $1≤i≤n$

I have already proved that the map $x\mapsto \text{Card}(\rho^{-1}(\{x\})$ is continuous on $X$ but i don't know how to continue ... I also thought of a proof by induction on $n=\text{sup}\{\text{Card}(\rho^{-1}(\{x\}),\: x\in X)\}$$n=\text{sup}\{\text{Card}(\rho^{-1}(\{x\})),\: x\in X\}$ which is clear for $n=1$.

Thank you for you help

Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that :

  • $X=\sqcup_{i=1}^{n}U_i$
  • the map $x\in X \mapsto \text{Card}(\rho^{-1}(\{x\})$ is constant on $U_i$ for all $1≤i≤n$

I have already proved that the map $x\mapsto \text{Card}(\rho^{-1}(\{x\})$ is continuous on $X$ but i don't know how to continue ... I also thought of a proof by induction on $n=\text{sup}\{\text{Card}(\rho^{-1}(\{x\}),\: x\in X)\}$ which is clear for $n=1$.

Thank you for you help

Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that :

  • $X=\sqcup_{i=1}^{n}U_i$
  • the map $x\in X \mapsto \text{Card}(\rho^{-1}(\{x\})$ is constant on $U_i$ for all $1≤i≤n$

I have already proved that the map $x\mapsto \text{Card}(\rho^{-1}(\{x\})$ is continuous on $X$ but i don't know how to continue ... I also thought of a proof by induction on $n=\text{sup}\{\text{Card}(\rho^{-1}(\{x\})),\: x\in X\}$ which is clear for $n=1$.

Thank you for you help

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MacFly
MacFly
  • 53
  • 4

Disjoint union of clopen sets such that the fibers has constant cardinality

Let $Z$ a compact set and $X$ a locally compact set. Let $p:Z\to X$ a local homeomorphism. Show that there exists $n≥1$ and $U_1,…,U_n$ open and closed sets of $X$ such that :

  • $X=\sqcup_{i=1}^{n}U_i$
  • the map $x\in X \mapsto \text{Card}(\rho^{-1}(\{x\})$ is constant on $U_i$ for all $1≤i≤n$

I have already proved that the map $x\mapsto \text{Card}(\rho^{-1}(\{x\})$ is continuous on $X$ but i don't know how to continue ... I also thought of a proof by induction on $n=\text{sup}\{\text{Card}(\rho^{-1}(\{x\}),\: x\in X)\}$ which is clear for $n=1$.

Thank you for you help